Ensemble Averaged Results

Ensemble averaging is carried out by calculating many signals from surfaces pos-sessing the same σ and λ0 values and averaging their result. This produces the coherent constituent of the reflected signal, signifying on average what the most likely reflected response would be from such a surface. Subsequently subtracting this coherent constituent from an individual scattered response provides the diffuse

4. Thickness Monitoring and Surface Roughness Detection

0 1 2 3 4 5 6 7 8 9 10

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

time (^s)

amplitude (Pa)

flat backwall rough backwall coherent diffuse

Figure 4.4: Example rough backwall simulated signal, showing the coherent component calculated using 500 simulated signals and the diffuse component.

component of the reflected signal. In general as roughness increases it is expected that coherency should reduce, while diffuse components which scatter from many points on the surface should increase depending on the position of the observer. A comparison of these signal constituents for a time domain pulse is shown in figure 4.4 for a surface with an RMS height of 0.2mm (σ = λ/8) and a correlation length of 0.8mm (λ0=λ/2).

Many (if not all) authors who have published work investigating the effects of sur-face roughness use ensemble averaging to damp out the large variations in amplitude and phase that can occur, thereby producing clear conclusions on the most prob-able trends that might be observed under given conditions. In a practical sense these results cannot be utilised without the possibility of averaging over a given surface, usually spatially by scanning a probe. In permanently installed monitoring situations this is clearly impractical where signals are only available from a single location. In this case a temporal average could be taken; however, the surfaces are not independent through time, therefore features developing on the surface (assum-ing gradual change) will affect the signals in a similar way, not allow(assum-ing ensemble averaging to take place. Accounting for this lack of applicability, ensemble aver-aged results can give insight into expected changes to signals and therefore will be examined briefly.

4. Thickness Monitoring and Surface Roughness Detection

Average Frequency Spectrum of Diffuse Signal, ₀=/2

=0.04

Frequency Spectrum of Coherent Signal, ₀=/2

=0.04

=0.09

=0.14

=0.19

(a) (b)

Figure 4.5: (a) Frequency spectra of coherent pulses reflected from rough surfaces with correlation lengths of 0.8mm (λ0 =λ/2) and RMS heights of 0.06, 0.14, 0.22 and 0.3mm (σ = 0.04λ, 0.09λ, 0.14λand0.19λ). (b) Average frequency spectra of diffuse components from the same surfaces.

The coherent pulse is first calculated as the average of all 500 simulated signals from rough surfaces with a particular RMS height and correlation length. Figure 4.5 (a) shows how the frequency spectrum of the coherent pulse changes as RMS height increases for a correlation length of 0.8mm (λ0 =λ/2). The diffuse component can then be extracted by subtracting the coherent pulse from each scattered pulse. Fig.

4.5 (b) shows the average frequency spectrum of the diffuse components from all 500 simulated results. The peak of the coherent signal shifts towards lower frequencies because the higher frequencies are preferentially scattered leaving only low ampli-tude, low frequency components. The peak of the diffuse component on average decreases slightly as lower frequency components start to get affected by increasing RMS height; however, it is clear that under these conditions all components are scattered to a similar extent indicating that frequency components within a single signal could not be used as indicators of the level of roughness.

Figure 4.6 (a) compares results from other correlation lengths (from λ0 = λ/4 to λ0 = 2λ/3), showing that the peak frequency in the coherent pulse is very similar below a σ/λ value of around 0.14. Above this value the location of the peak frequency was found to vary; however this was due to very low coherent amplitudes affecting

4. Thickness Monitoring and Surface Roughness Detection

Figure 4.6: (a) Peak frequencies in frequency spectra of coherent pulse and average frequency spectra of diffuse components for all RMS height values (0.02 to 0.3mm in 0.02mm increments) and correlation length values (0.4, 0.8, 1.6 and 2.4mm) investigated.

(b) Amplitudes at peak frequencies.

peak location markedly. The peak frequency in the diffuse components are also very similar when λ0 ≥λ/2 regardless of σ; however below this value the peak doesn’t migrate as rapidly towards the lower frequencies which can be seen when λ0=λ/4.

This is most likely caused by longer wavelength (low frequency) components not interacting as strongly with the shorter features along the rough surface with low λ0 values.

Figure 4.6 (b) compares the amplitude at the peak within the frequency spectrums for the same simulated results. It can be seen that the coherent amplitude is signif-icantly higher when λ0 =λ/4. This is likely to be caused by more of the surface on average being orientated such that energy is directed towards the receiver with a low correlation length. With a long correlation length it’s more likely that a majority of the signal is directed away from the receiver be it diffuse or coherent, therefore

4. Thickness Monitoring and Surface Roughness Detection

over many surfaces less coherent energy is received. There could also be an effect of increased multiple scattering at low correlation lengths simply increasing the energy content of the received pulse. Looking at the diffuse component it can be observed that for short correlation lengths and high RMS heights where multiple scattering effects cannot be ignored (λ₀ < λ/2) the diffuse component of the scattered pulse is on average larger in amplitude than the scattered pulse under flat backwall con-ditions. In general the results in Fig .4.6 show that the frequency content of the coherent and diffuse components within the scattered pulse are much more sensitive to RMS height than correlation length. This is also illustrated by the transition from coherent energy to diffuse energy domination within the scattered pulses oc-curring consistently at RMS heights around 0.08λ for each of the correlation lengths considered.

The fact that higher frequency components of a pulse are preferentially scattered before lower frequency components also has a direct impact on the choice of the incidence pulse centre frequency. Higher frequency signals generally allow more pre-cise TOF calculations to be made because they have a sharper rising edge; however, roughness tends to increase diffuse energy content of a scattered signal as frequency (and therefore wavelength) decreases. The scope of this thesis is such that only the operating conditions of the sensor being used will be investigated further (∼2MHz centre frequency); however these two opposing requirements of high frequency for precision and low frequency for insensitivity to surface roughness can make definition of pulse centre frequency challenging.